Vacuum Rabi Oscillation

A vacuum Rabi oscillation is a damped oscillation of an initially excited atom coupled to an electromagnetic resonator or cavity in which the atom alternately emits photon(s) into a single-mode electromagnetic cavity and reabsorbs them. The atom interacts with a single-mode field confined to a limited volume ''V'' in an optical cavity.

Spontaneous emission is a consequence of coupling between the atom and the vacuum fluctuations of the cavity field.

Mathematical treatment

A mathematical description of vacuum Rabi oscillation begins with the Jaynes–Cummings model, which describes the interaction between a single mode of a quantized field and a two level system inside an optical cavity. The Hamiltonian for this model in the rotating wave approximation is

:$\hat_ = \hbar \omega \hat^\hat +\hbar \omega_0 \frac +\hbar g \left(\hat\hat_+ +\hat^\hat_-\right)$

where $\hat$ is the Pauli z spin operator for the two eigenstates $, e \rangle$ and $, g\rangle$ of the isolated two level system separated in energy by $\hbar \omega_0$; $\hat_+ = , e \rangle \langle g ,$ and $\hat_- = , g \rangle \langle e ,$ are the raising and lowering operators of the two level system; $\hat^$ and $\hat$ are the creation and annihilation operators for photons of energy $\hbar \omega$ in the cavity mode; and

:$g=\frac\sqrt$

is the strength of the coupling between the dipole moment $\mathbf$ of the two level system and the cavity mode with volume $V$ and electric field polarized along $\hat$.

The energy eigenvalues and eigenstates for this model are
:$E_(n) = \hbar\omega \left(n+\frac\right) \pm \frac \sqrt=\hbar \omega_n^\pm$

:$, n,+\rangle= \cos \left(\theta_n\right), g,n+1\rangle+\sin \left(\theta_n\right), e,n\rangle$

:$, n,-\rangle= \sin \left(\theta_n\right), g,n+1\rangle-\cos \left(\theta_n\right), e,n\rangle$

where $\delta = \omega_a - \omega$ is the detuning, and the angle $\theta_n$ is defined as

:$\theta_n = \tan^\left(\frac\right).$

Given the eigenstates of the system, the time evolution operator can be written down in the form

:$\begin e^ & = \sum_ \sum_ , n,\pm \rangle \langle n,\pm, e^ , n',\pm \rangle \langle n',\pm, \\ &= ~e^ , g,0\rangle \langle g,0, \\ & ~~~+ \sum_^\infty \\ & ~~~+ \sum_^\infty \\ \end.$

If the system starts in the state $, g,n+1\rangle$, where the atom is in the ground state of the two level system and there are $n+1$ photons in the cavity mode, the application of the time evolution operator yields

:\begin
e^ , g,n+1\rangle
&= (e^(\cos^2, g,n+1\rangle+\sin\cos, e,n\rangle)
+ e^ (-\sin^2, g,n+1\rangle-\sin\cos, e,n\rangle)\\
&= (e^+e^) \cos, g,n+1\rangle
+ (e^-e^) \sin, e,n\rangle\\
&= e^\Biggr cos_\biggr[\frac\biggrg,n+1\rangle
+_\sin\biggr[\frac\biggr.html" ;"title="frac\biggr.html" ;"title="cos \biggr[\frac\biggr">cos \biggr[\frac\biggrg,n+1\rangle
+ \sin\biggr[\frac\biggr">frac\biggr.html" ;"title="cos \biggr[\frac\biggr">cos \biggr[\frac\biggrg,n+1\rangle
+ \sin\biggr[\frac\biggre,n\rangle\Biggr]
\end.

The probability that the two level system is in the excited state $, e,n\rangle$ as a function of time $t$ is then
: \begin
P_e(t) & =, \langle e,n, e^ , g,n+1\rangle , ^2\\
&= \sin^2\biggr frac\biggr\
&= \frac \sin^2
\end

where $\Omega_n=\sqrt$ is identified as the Rabi frequency. For the case that there is no electric field in the cavity, that is, the photon number $n$ is zero, the Rabi frequency becomes $\Omega_0=\sqrt$. Then, the probability that the two level system goes from its ground state to its excited state as a function of time $t$ is

:$P_e(t) =\frac \sin^2$

For a cavity that admits a single mode perfectly resonant with the energy difference between the two energy levels, the detuning $\delta$ vanishes, and $P_e(t)$ becomes a squared sinusoid with unit amplitude and period $\frac.$

Generalization to ''N'' atoms

The situation in which $N$ two level systems are present in a single-mode cavity is described by the Tavis–Cummings model

, which has Hamiltonian

:$\hat_ = \hbar \omega \hat^\hat +\sum_^N.$

Under the assumption that all two level systems have equal individual coupling strength $g$ to the field, the ensemble as a whole will have enhanced coupling strength $g_N=g\sqrt$. As a result, the vacuum Rabi splitting is correspondingly enhanced by a factor of $\sqrt$.

See also

* Jaynes–Cummings model
* Quantum fluctuation
* Rabi cycle
* Rabi frequency
* Rabi problem
* Spontaneous emission

References and notes

{{reflist

Quantum optics
Atomic physics
Atomic, molecular, and optical physics